Solve the Polynomial Equation: 16x² + x³ = 0 Step by Step

Shown below is the given equation:

$16x^2+x^3=0$

First, note that on the left side we are able to factor the expression by using a common factor. The largest common factor for the numbers and letters in this case is $x^2$due to the fact that the square power (second) is the lowest power in the equation and therefore is included both in the term with the third power and in the term with the second power. Any power higher than this is not included in the term with the square power, which is the lowest. Therefore this is the term with the highest power that can be factored out as a common factor from all terms in the expression. Continue to factor the expression:

$16x^2+x^3=0 \\ \downarrow\\ x^2(16+x)=0$

Proceed to the left side of the equation that we obtained in the last step. There is a multiplication of algebraic expressions and on the right side the number 0. Therefore given that the only way to obtain 0 from a multiplication is to multiply by 0, at least one of the expressions in the multiplication on the left side must equal zero,

meaning:

$x^2=0 \hspace{8pt}\text{/}\sqrt{\hspace{6pt}}\\ x=\pm0\\ \boxed{x=0}$

(In this case, taking the even root of the right side of the equation will yield two possibilities - positive and negative, however since we're dealing with zero, we only obtain one solution)

Or:

$16+x=0\\ \downarrow\\ \boxed{x=-16}$

Let's summarize the solution of the equation:

$16x^2+x^3=0 \\ \downarrow\\ x^2(16+x)=0 \\ \downarrow\\ x^2=0 \rightarrow\boxed{ x=0}\\ 16+x=0 \rightarrow \boxed{x=-16}\\ \downarrow\\ \boxed{x=0,-16}$

Therefore the correct answer is answer B.